Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents

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Introduction
This paper is devoted to studying qualitative properties of nonnegative weak solutions for the following doubly nonlinear parabolic problem with variable exponents where Ω is a bounded domain of R  ,  ≥ 1, with smooth boundary Ω, (, ) = || ()−1  and Δ (,)  is defined as Δ (,)  = div (|∇| (,)−2 ∇) . ( The exponents , ,  and the coefficient  are given measurable functions.It will be assumed throughout the paper that these functions satisfy some specific conditions. Problems of this form appear in various applications; for instance in models for gas or fluid flow in porous media ( [1,2]) and for the spread of certain biological populations ( [3]).Our motivation to study problem P with variable exponents is the fact that it is considered as a model of an important class of non-Newtonian fluids which are well known as electrorheological fluids, see ([4]).It appears also as a model in image restoration ( [5]) and in elasticity ( [6]).
It is well known that solutions of problems such as P exhibit various qualitative properties, which reflect natural phenomena, according to certain conditions on (, ), (, ), (), (, ), and  0 , (see for example [7][8][9][10][11][12][13] and the references therein).Among the phenomena that interest us in this work is the finite speed of propagation, which means that if  0 > 0 is such that supp( 0 ) ⊂ ( 0 ,  0 ), then supp((, )) ⊂ ( 0 , ()), for any  ∈ (0,), where () is a positive function which depends on  0 , (i.e., solutions with compact support).This property has various physical meanings; for instance, in the study of turbulent filtration of gas through porous media, a solution with compact support means that gas will remain confined to a bounded region of space, (see [14]).
The phenomenon of finite speed of propagation was investigated by Kalashnikov in [15].He considered, for  = 1, the equation ()/ − Δ = 0 in R × (0, ∞) and, under specific conditions, proved that if the initial condition  0 has a compact support, then the condition ∫ 0 + (1/()) < +∞ is necessary and sufficient for solutions to have compact support.This result was extended by Dìaz for  ≥ 1, in [16].Later, in [17] Dìaz and Hernández considered the doubly nonlinear problem with absorption term ()/ − Δ   + || −1  = 0, in R  × (0, ∞), where () = || −1 .Under the assumption that  0 has a compact support and 0 <  <  − 1, they proved that any solution has a compact support for all  > 0. This result was obtained by the construction of a local uniform super-solution.Let us recall that the finite speed of propagation phenomenon has been studied by many authors in the last decades, (see [18][19][20][21]).
Otherwise, it is worth noting that problem P has been treated by Antontsev and Shamarev in several papers.In [29,30], they proved the existence of weak and strong solutions.Moreover, under certain regularity hypotheses on (), (, ), and under the sign condition (, ) ≤ 0 a.e, they studied properties of finite speed of propagation and extinction in finite time in [9,10].Their results were established by using the local energy method.Here, we shall use the so-called method of sub-and supersolutions to extend some of the results in [9,10].To the best of our knowledge, there are few results concerning the study of qualitative properties for parabolic equations with variable exponents by using this method.Furthermore, we shall also extend to the parabolic case some of the results by Zhang et al. in [31], where radial sub-and supersolutions for some elliptic problems with variable exponents are constructed, and some of the results by Chung and Park in [22] and by Yuan et al. in [27], to variable exponents case.In fact, we shall exploit their arguments in our parabolic problem setting with less conditions on the exponents (, ), (, ), and () and the coefficient (, ).
The present paper is organized as follows.In Section 2, we introduce some basic facts about the variable exponents spaces.In Section 3, we give assumptions and general definitions; then, we establish a comparison principle which ensures the uniqueness of solutions.In Section 4, we investigate the extinction and nonextinction properties for the solution of P. Finally in Section 5, we study the property of finite speed of propagation.
A function  is a weak solution of P if it is simultaneously a supersolution and a subsolution.
The following result concerning the local existence of weak solutions of problem P is established in [29].
Theorem 5. Let  ∈  0 (Ω), (, ) satisfies the log-Hölder condition in   (14), and let conditions (20) and ( 21) be fulfilled.Moreover, we assume that and the exponents ,  satisfy one of the following conditions (1)  is independent of , and () > 0 in Ω, Then, the problem P has at least one nonnegative weak solution in   * , with Moreover, for small  the solution satisfies the estimate with a constant  depending only on the data.
The following comparison principle is essential to prove uniqueness and qualitative properties of nonnegative solutions.Proposition 6.Let  (respectively V) be a subsolution (respectively supersolution) of P, with the initial datum  0 (respectively V 0 ), satisfying (21).We assume that (/)(, ), (/)(, V) ∈  1 (  ), and that conditions (20) are fulfilled.If either (, ) ≤ 0 a.e. in   , or  + ≤  − , then we have  ≤ V a.e. in   .Remark 7. Note that the comparison principle is true for weak solutions  with (/)(, ) ∈  1 (  ) ∩   (  ) and recall that in the papers [29,30], the authors gave some conditions on the data of problem P in order to ensure that this class of solutions is nonempty.
Proof.We consider the test function and  > 0 is small.It is easy to see that where sign + () = 1, if  > 0, and sign + () = 0, if  ≤ 0.Moreover, we claim that for all , V ∈ (  ) the function   ( − V) ∈ (  ).Indeed, we observe that for all  ∈ R, |  ()| ≤ 1.Then, by Proposition 2 On the other hand, we have Hence, from Proposition 2 we get Therefore, combining (29) and (31) we deduce the claim.On the other hand, from Definition 4, we obtain Due to a monotonicity argument, we have then Then, from (34) and (35), by letting  → 0, we obtain Hence, if (, ) ≤ 0 a.e. in   , it follows that Then, by Gronwall's lemma we deduce the desired result.Now, we continue the proof without any sign condition on (, ).From (37), by using  + ≤  − and the Lebesgue's dominated convergence theorem it follows that where  is depending on the supnorms of  and V. Hence we deduce from Gronwall's lemma that

Finite Time Extinction and Nonextinction
This section is devoted to studying extinction and positivity properties for nonnegative solutions of problem P, without any sign condition on the coefficient (, ), and according to the ranges of (, ), (, ), and ().The proof of the results is based on the construction of suitable sub-and supersolutions and on the use of the preceding comparison principle given in Proposition 6. Proof.We consider the following function

Finite Time
where and where  > 0 will be specified later.Our goal is to prove that V is a supersolution of P and by comparison principle, we can thus deduce the result.Firstly, we shall show that For all  ∈ Ω and  > 0, we have and which implies that V ∈  ∞ (  ) ∩ (  ).Moreover, we have and hence (/)V () ∈  ∞ (  ).Due to the embedding we get that (/)V () ∈   (  ).
Next, we will mention an extinction result where there is no condition between the ranges of (, ) and ().Proposition 10.Let  be a strong solution of P. Assume that  − >  + and (, ) ≤ − < 0, sup   |∇(, )| < ∞ and ‖ 0 ‖ ∞ is small enough.Then, there exists a finite time  1 such that for all  ≥  1  (, ) = 0, .. ∈ Ω. (64) Proof.We consider the same supersolution V(, ) as in the proof of Theorem 9 but we choose here  = 1, which means where and We have already shown in Theorem 9 that −Δ (,) V ≥ 0. We claim that   V () ≥  (, )  (,) , (69) by using the same lines as in the proof of Theorem 9. Since  + <  − , it is therefore sufficient to have which is satisfied if we choose Consequently, by the comparison principle we deduce the extinction of solution in finite time.

Nonextinction of
The method of proof is inspired from [27], where the constant exponents case is treated.However, some difficulties arise in the construction of subsolutions due to the fact that the exponents are variable.The proof of this theorem is divided into two lemmas.In the first lemma, we show by using a comparison function that the support of weak solution is nondecreasing with respect to time.In the second lemma, we show that the solution is positive locally in Ω; then, by a finite covering argument, we deduce the result.
The proof of Lemma 12 follows the same lines as that of lemma 4.2 in [22], where the constant exponents case is studied.For completeness, we shall give it here.
Proof.The argument used here is based on a comparison function with which we show that the support of solution is increasing.For that we consider  an arbitrary set which is a nonzero measure subset of Ω such that inf ∈  0 ̸ = 0. We divide the proof in two cases, firstly we treat the case where  − ≥ 1 and then the case where  + < 1.If  − ≥ 1, we consider the following function: where , and it is easy to verify that On the other hand, by direct calculations we get Hence Since () ≥ 1 and V 1 (, ) ≤ 1 for all  ∈ ,  > 0, it follows that Moreover, from the definition of V 1 , we have V 1 (, 0) ≤  0 , almost everywhere in , and V 1 (, ) = 0 for all,  ∈ ,  > 0. Thus, by comparison principle we conclude that for any arbitrary  where inf ∈  0 > 0, the weak solution of P satisfies  (, ) > 0 a.e. ∈ , and all  > 0; (79) and the result follows in this case.If  + < 1, we consider the following function: where  = min{inf ∈  0 , 1}.By the same argument used previously we obtain that V 2 ∈  ∞ (  ) ∩ (  ), and (/)V () 2 ∈   (  ).Moreover, by direct calculations we get since  + < 1 and V 2 (, ) ≤ 1 for all  ∈ ,  > 0. Hence Therefore, we have (83) Thus, by the same argument used previously we deduce our result.
Lemma 13.Under the same assumptions of Theorem 11, let the initial condition satisfies inf ∈  ( 0 )  0 ̸ = 0, for some 0 <  <  ≤ 1/2.Then, there exists   > 0, such that for any  ≥   , where Proof.We consider the following function: where and where  and  are positive constants small and large enough, respectively,  is a positive constant such that and , ,  are positive constants and will be determined later.By direct calculations, we get and    () = − ()  +()  +()  −()/−1 () .
Therefore, the result follows from Lemma 12.
Proof of Theorem 11.The proof is similar to that of Theorem 1.2, in [27], and we omit the details here.

Finite Speed of Propagation Property
In this section we shall give precise estimates for the of support of the solution of P, depending on the size of the support of  0 .Let us emphasize that each estimation is obtained under a sign condition on (, ) and depending on the range of the exponents (, ), (, ), and ().
As in [21], the proof is based on the construction of local supersolutions and on the use of the comparison principle.
Concerning the construction of supersolutions, we shall proceed as in [31].
Combining the same lines as in the previous theorem and the comparison principle in Proposition 6, we conclude the result.
Finally, we state the following result on uniform localization of the support of solution.

Remark 17.
As in [20,21], we can assume in this section that Ω is a set of R N , not necessarily bounded.